Let C be the circle radius 1, center the origin. A point P is chosen at random on the circumference of C, and another point Q is chosen at random in the interior of C. What is the probability that the rectangle with diagonal PQ, and sides parallel to the x-axis and y-axis, lies entirely inside (or on) C?
Given P, let R be the rectangle inscribed in C with sides parallel to the axes. Then the rectangle with diagonal PQ lies entirely inside C iff Q lies inside R.
Let O be the center of C. Let the diameter of C parallel to the y-axis make an angle θ with OP. Then the area of R is 4 sin θ cos θ. So the required probability is 2/π ∫0π/2 (4 sin θ cos θ)/π dθ = 4/π2 ∫0π/2 sin 2θ dθ = 4/π2.
46th Putnam 1985
© John Scholes
7 Jan 2001